Kraichnan's energy-enstrophy theory for 2D
inviscid flows on the sphere is discussed within a variational
framework. We will give necessary and sufficient conditions for
the existence and uniqueness for the extremals of the energy with
zero circulation under different values of the temperature
parameter $\beta$. The unboundedness of the augmented energy
functional in this model when $\beta$ is located in the certain
intervals will be shown and related to energy catastrophe of the
energy-enstrophy model.